Late-time draining of a thin liquid film on the outer surface of a circular cylinder

A combination of analytical and numerical techniques is used to give a complete description of the late-time draining of a two-dimensional thin liquid film on the outer surface of a stationary horizontal circular cylinder. In this limit three regions of qualitatively different behavior emerge, namely a draining region on the upper part of the cylinder anda pendantdrop region on the lower part of the cylinder joined by a narrow inner region. In the draining region, capillarity is negligible and the film thins due to gravity. In the pendant-drop region (which, to leading order, contains all of the liquid initially on the cylinder), there is a quasistatic balance between gravity and capillarity. The matching between the draining and pendant-drop regions occurs via the inner region in which the film has a capillary-ripple structure consisting of an infinite sequence of alternating dimples and ridges. Gravity is negligible in the dimples, which are all thinner than the film in the draining region. On the other hand, gravity and capillarity are comparable in the ridges, which are all thicker than the film in the draining region. The dimples and the ridges are all asymmetric: specifically, the leading-order thickness of the dimples grows quadratically in the downstream direction but linearly in the upstream direction, whereas the leading-order film thickness in the ridges goes to zero linearly in the downstream direction but quadratically in the upstream direction. The dimples and ridges become apparent in turn as the draining proceeds, and only the first few dimples and ridges are likely to be discernible for large but finite times. However, there is likely to be a considerable period of time during which the present asymptotic solution provides a good description of the flow.

Automated summary

A combination of analytical and numerical techniques is used to provide a detailed description of the late-time draining behavior of a thin liquid film on a stationary horizontal circular cylinder. The film can be divided into draining and pendant-drop regions, connected by a narrow inner region. The draining region is characterized by thinning of the film due to gravity, while the pendant-drop region exhibits a quasistatic balance between gravity and capillarity. These two regions are connected through an inner region with a capillary-ripple structure. The dimples and ridges in the inner region have different thickness profiles and become observable as the draining progresses.